I am cross-posting this question which I posted in math.stackexchange.com since I realized that there are people in Mathoverflow who are not signed-up there.
Edit: For a topological group $G$ and a topological $G$-module $M$, we define the $n$th group of continuous cochains $C^n = C^n(G,M)$ as the group of continuous maps $G^n \rightarrow M$. We define a coboundary map $d_n : C^n \rightarrow C^{n+1}$ by the formula: $$ (d_n f)(g_1, \ldots, g_{n+1}) = g_1 f(g_2, \ldots, g_{n+1}) + \sum_{i=1}^n (-1)^i f(g_1, \ldots, g_ig_{i+1}, \ldots, g_{n+1}) + (-1)^{n+1} f(g_1, \ldots, g_n). $$
This yields a complex $C^{\bullet}(G,M)$. We define the $n$th cohomology group of $G$ with coefficients in $M$ by $H^n(G,M) := \text{ker } d_n / \text{im } d_{n-1}$.
Now, let $K$ be a finite extension of $\mathbb{Q}_p$ and $K^s$ a separable closure of $K$. Put $G_K:=Gal(K^s/K)$. Let $V$ be a finite-dimensional $\mathbb{Q}_p$-vector space with a continous action of $G_K$ given by $ρ:G_K \rightarrow GL(V)$.
My question is: Is there a difference between the cohomology groups $H^n(G_K,V)$ and $H^n(G_V,V)$, where $G_V=ρ(G_K)$?
I am guessing that if there is such a difference, it might have something to do with their respective topologies, (profinite topology on $G_K$ and $p$-adic topology on $GL(V)$) but I am not really sure.